Trigonometry , like most abstract sciences, has both an operational side and a quantitative side.

The operational side is dominated by the hypothesis that a function f(x) exists which has the property

It is from this that the usual rules for operating with the trigonometric functions are deduced.

In 1877, the English mathematician J J. Sylvester (1) announced the discovery of a rather complete formal analogy between the mathematical theory of the invariants and covariants of binary quantics and the chemical theory of valency as it then stood.

When exploring some triangle geometry, we stumbled upon a set of Cevian axes: lines having certain properties in common with the Euler line. These axes yielded interesting results that we had not seen before. However, we have since learnt that many of them were discovered in the 19th Century. We were also unaware that John Rigby had covered much of this ground in unpublished work in the 1990s (private communication). We are therefore reluctant to claim any of our results as new, even though we hope that some may be.

Scout‘s pace, according to my old boy scout manual, consisted of alternately walking and running 22 yards. I believe it is also used by military scouts and messengers who wish to travel at a moderate speed without too much effort. As I have also discovered, it is useful for elderly, overweight athletes with a low pain tolerance.

Some of my distinguished predecessors have ably dealt in their Presidential addresses with questions that are most closely related to the aims of our Association, namely, the teaching of Mathematics in Schools. Others have sought to speak on subjects in which they were actively interested. I have followed this precept and accordingly I propose to address you upon a subject closely related to my own scientific interests, namely, the exploration of our universe. Naturally, I cannot claim to speak with authority over its whole range.

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About 75 B.C., Cicero the orator, stumbled upon an ancient monument. The lettering was illegible, but ‘on clearing away the brambles and thorns’ he discovered a curious design—a sphere with circumscribed cylinder.

Nothing can convince us more that “things are not what they seem” than a comparison of the descriptions by two men of what they remember of Cambridge in the seventies ! I should of course recognise my friend Professor Forsyths picture of the Cambridge of that time and am grateful for his calling to my mind so many memories of the past. But for me he has painted them in far too drab a colour, so that one wonders whether his undergraduate days could have been like mine some of the happiest in his life. The reading of his paper enforces upon me the truth of the principle that environment and personalities are not things in themselves, but are our own “constructs”—each of us giving our own “reality” to such evasive phenomena.

Theorem. If a he any positive quantity not equal to 1, and x, y, z be any three rational quantities in descending order of magnitude, then

“A derivative is a linear transformation which approximates a function near to a point.” I heard this definition for the first time in 1964, in a lecture at Oxford on the calculus of several variables (the lecturer was D.A. Edwards). The words came as a revelation, and from then on I had a new conception of differential calculus. At school, I thought that I knew about calculus because I could write out proofs of differentiation ‘from first principles’. I enjoyed going through the hoops and proving that δy/δ tends to a limit, but somehow the idea of differentiation never became ‘part of me’.